In the case of quadratic forms, there is an interpretation of the determinant in terms of Galois cohomology, which is given as follows:
Let (V, b) be a quadratic form of dimension n over K. Since every quadratic form is diagonal, there is an isomorphism (V, b)K¯ ∼= ( ¯Kn,h1, . . . ,1i2) of quadratic forms over the separable closure ¯K. The automorphism group of the quadratic form ( ¯Kn,h1, . . . ,1i2) is the orthogonal group On. By Weil descent, this gives a bijection between the set of isomorphism classes of quadratic forms of dimensionn overK and the cohomology set H1(K,On). Note that On is not an abelian group, so that we are dealing with non-abelian Galois cohomology, andH1 is not a group, but a pointed set. For a detailed exposition on non-abelian group cohomology, the reader may refer to ([29], Chap. I, §5).
For n ∈ N, the determinant morphism det : On → µ2 induces a coho-mology map det : H1(K,On) → H1(K, µ2). Identifying quadratic forms with the elements of cohomology sets H1(K, µ2) via Weil descent and H1(K, µ2) with K∗/K∗2 via the Kummer isomorphism, this map coincides with the determinant det : ˆW(K)→K∗/K∗2 (cf. [21],§2.4).
We want to apply this technique tor-forms, so we shall recall how the bijections obtained by Weil descent are explicitly given. The following Lemma is cited from ([30], Chap. X, Prop. 2.4):
4.1 Lemma. (Weil Descent)
Let L/K be a Galois field extension and let Ψ∈Wˆr(L) be an r-form over L. Let E(L/K,Ψ) be the set of r-forms over K which become isomorphic to Ψ over L.
For Θ ∈ E(L/K,Ψ), let f : ΘL ∼
→ Ψ be an isomorphism over L. Then the map a =a(Θ) : G(L/K) →AutL(Ψ) given by s 7→ as :=f sf−1 is a 1-cocycle and the assignment Θ7→a induces a bijection
E(L/K,Ψ)↔H1(L/K,AutL(Ψ)).
We introduce some notation: Let n ∈ N, and let A be a set. We write the elements of then-fold direct sumA⊕nas vectors Σiaiei withai ∈A. The left action of the n-th symmetric groupSn is denoted by σ(Σiaiei) := Σiaieσi for σ ∈Sn and Σiaiei ∈A⊕n.
4.2 Definition. Let A be a group. The wreath product Sn∫A of Sn with A is defined as the set Sn ×A⊕n with the semidirect product, which is defined as (σ, a)·(τ, b) = (στ, aσb) for σ, τ ∈Sn and a, b∈A⊕n.
We have a short exact sequence of groups
1 // A⊕n // Sn∫A x // Sn //1 (1)
with a natural splitting given byσ 7→(σ,1). In the case thatA⊂R∗is a subgroup of the multiplicative group of some ring R, we identify Sn∫A with a subgroup of GLn(R) by the embedding
Sn∫A ,→ GLn(R), (σ,Σiaiei) 7→ (δi,σj ai)i,j. where δi,j is the Kronecker symbol.
Now let G be a group and let A be a multiplicative G-group. The follow-ing notations are taken from Serre’s exposition on non-abelian group cohomology in ([29], Chap. I, §5): Consider Sn as a G-group with trivial G-action, and let ρ ∈ Hom(G, Sn) be a 1-cocycle. Then ρ induces a twisted G-action on A⊕n by
s0(Σiaiei) = Σiaieρsi (cf. [29], Chap. I,§5.3). ThisG-module is denoted byρ(A⊕n).
We get a short exact sequence of twisted G-groups
1 // ρ(A⊕n) // ρ(Sn∫ A) x // ρ(Sn) //1,
where ρ(Sn) is the set Sn with G-action given by s0σ =ρsσρs−1 for s ∈ G, σ ∈ Sn. The cocycle ρ ∈ Hom(G, Sn) describes a G-action on the set {1, . . . , n}. Let n = Σknk be the orbit decomposition under this action. This induces decomposi-tions ρ=L
kρk and ρ(A⊕n) =L
k ρk(A⊕nk) with ρk∈Hom(G, Snk). We say that ρ is transitive if it describes a transitive G-action on the set of n elements, i.e. if k = 1 in the above notation.
The following lemma gives a classification of separable K-algebras by Galois cohomology with values in the symmetric group:
4.3 Lemma. Let G := G( ¯K/K) be the absolute Galois group of the field K, and let Sn be the trivial G-module. Then there is a bijection between the coho-mology set H1(K, Sn) and the set of equivalence classes of separable K-algebras of dimension n modulo K-algebra-isomorphism, given as follows: For a transi-tive cocycle ρ∈ Hom(G, Sn), let L(ρ)⊂ K¯ be the field fixed by the stabilisator of some point in {1, . . . , n}. For arbitrary ρ with orbit decomposition ρ =L
kρk let L(ρ) = L
kL(ρk). The inverse map is given by assigning a separable field exten-sion L/K of dimension n to the cocycle corresponding to the action of GK on the set GK/GL.
Proof: This can be seen as an application of Weil Descent in a more general setting that the one given in Lemma 4.1. In this situation, however, one may simply check
that the given maps are inverse bijections.
4.4 Lemma. LetV be an indecomposabler-form overK. ThenVK¯ ∼=W⊕· · ·⊕W with an indecomposable r-form W over K.¯ V is separable if and only if W is 1-dimensional over K.¯
Proof: LetVK¯ ∼=Ls
j=1V˜j⊕mj be the decomposition into pairwise inequivalent inde-composabler-forms ˜Vj over ¯K. By Lemma 4.1, Aut(VK¯) =L
jAut( ˜V⊕mj) induces a decompositionV =Ls
j=1Vj ofr-forms overK . ButV is indecomposable, hence s= 1.
Remark: In the special case that (V,Θ) is separable, this was already proved in
Lemma 3.8(ii).
4.5 Lemma. Letr≥3and let(V,Θ) be an indecomposable r-space overK. Then Aut (V⊕n,Θ⊕n) =Sn∫Aut (V,Θ) .
In particular, the automorphism group of a diagonal r-form of dimension n over K¯ is the wreath product Sn∫ µr.
Proof: Let A ∈ Aut(V⊕n,Θ⊕n). Write V⊕n = V e1 ⊕ · · · ⊕V en and assume that A =P
i,jAi,j with Ai,j ∈Hom(V ei, V ej) ∼= End(V). We will show that for every j ∈ {1, . . . , n} there is k = k(j) ∈ {1, . . . , n} such that Ai,j = 0 for i 6= k. Since A is an automorphism, this implies that j 7→ k(j) is a permutation and that Ak(j),j ∈ Aut(V) for j = 1, . . . , n. Then A =P
If Akj is invertible, then Aij = 0 since Θ is regular. Assume Akj is not invertible and let V0 := im(Akj), V00 = L
i6=kim(Aij). Then the restriction of A gives an isomorphism of r-forms
(V,Θ)∼= (V ej,Θ⊕n|V ej)→∼ (V0,Θ⊕n|V0)⊕(V00,Θ⊕n|V00) .
This contradicts the assumption that (V,Θ) is indecomposable.
For A∈GLn(K), B ∈GLn(K), let
Together with the inclusions Sn∫µr ⊂GLn( ¯K), this addition and multiplication induce the structure of a semiring on the set S
n∈N
H1(K, Sn∫µr). For this we have 4.6 Theorem. Descent gives an isomorphism of semirings
Wˆr+sep(K)→∼ [
n∈N
H1(K, Sn∫ µr).
Proof: Immediate from the previous lemmas.
4.7 Lemma. Let (L,trL/Khbir) be an indecomposable separable r-form of degree n over K. Let t1, . . . , tn ∈ GK be a set of representatives for GK/GL and choose β ∈K¯ with βr =b. Let ρ∈ Hom(GK, Sn) such that sti ∈ tρsiGL for s ∈GK and i = 1, . . . , n, or equivalently, such that ρ corresponds to L/K under the bijection in Lemma 4.3. Then the image of (L,trL/Khbi) under the bijection in Lemma 4.6 4.8 Theorem. Via the identifications in Lemma 4.3 and Theorem 4.6, the sur-jective map
Proof: The projection splits, hence the center map is surjective. From Lemma 3.8, we know that every indecomposable r-form over K with center L is isomorphic to (L,trL/Khbir) for some b ∈ L∗. Lemma 4.7 shows that the cocycle given by
projection to Sn is just the one describing L.
In order to classify separable r-forms using Galois cohomology, we need to compute Galois cohomology of ρ(Sn) and ρ(µ⊕rn), where Galois action is twisted
4.9 Lemma. (Hilbert 90)
Let M/K be a Galois extension, and let ρ ∈ Hom(GK, Sn). Then we have H1(GM/K,ρ(M∗⊕n)) = 1.
Proof: Using direct sum decomposition, we may assume thatρis transitive. First, letM/Kbe finite. WriteG=GM/K. As in the classical proof (cf. [30], Chap. X,§1, extension, take the direct limit over its finite subextensions.
4.10 Lemma. Letρ∈Hom(GK, Sn)be transitive and letL/K be the of those permutations which commute with the action of GK on HomK(L,K¯).
The action of AutK(L) on HomK(L,K) by right translation gives an embedding¯ AutK(L)opp ,→ H0(GK,ρ(Sn)). In order to prove surjectivity of this map, let σ ∈H0(GK,ρ(Sn)). Then s(σ¯t1) = σ(s¯t1) = σ¯t1 for all s∈ GL, which shows that
σ¯t1 = ¯tσ1 ∈AutK(L). Furthermore, we have σ¯ti =σ(ti¯t1) =ti(σ¯t1) =titσ1 for all i= 1, . . . , n, which shows that σ is given by right translation with tσ1.
4.11 Lemma. Let L/K be a finite separable field extension. Then the isomor-phism classes of r-forms with center isomorphic to L correspond bijectively to the orbits in L∗/L∗r under the action of AutK(L).
Proof: This follows from Theorem 4.8, Lemma 4.10, and ([29], Chap. 1,§5.5, Cor. 2
to Prop. 39).
Remark. We have given a new proof for the classification of separabler-forms in Lemma 3.8(ii).
Now we want to use the cohomological classification for separable r-forms to define first degree cohomological invariants. For this purpose, we study group homomorphism on the wreath product:
4.12 Definition. LetMn(K)denote the space of n×n-matrices over the fieldK.
The permanent per :Mn(K)→K is defined by the modified Leibniz Formula per(aij) := P
σ∈Sn
n
Q
i=1
ai,σj.
In particular, for a= (σ,(α1, . . . , αn))∈Sn∫µr, we have per(a) =
n
Q
i=1
αi. 4.13 Lemma. Let r, n >0. Then
Hom(Sn∫ µr,K¯∗) = Z/(r)×Z/(2),
where Z/(r) is generated by the permanent per : Sn∫µr → µr, and Z/(2) is gen-erated by the sign sgn :Sn∫µr→Snsgn→µ2.
Proof: From the short exact sequence (1), we obtain the exact sequence of first terms
0 // Hom(Sn,K¯∗) x // Hom(Sn∫µr,K¯∗) // Hom(µnr,K¯∗)Sn.
The group Hom(Sn,K¯∗) is isomorphic toZ/2, generated by the sign, and the group Hom(µnr,K¯∗)Sn is isomorphic toZ/r, generated by the product Π. The right arrow splits by Π7→per, hence the sequence is a direct product.
4.14 Definition. (Cohomological Invariants for Separable r-Forms) We identify separable r-forms with elements of the cohomology sets H1(K, Sn∫ µr) via the bijection in Theorem 4.6 and H1(K, µr) = K∗/K∗r via the Kummer iso-morphism.
(i) Let per : ˆWrsep(K)→K∗/K∗r be the cohomology map per :H1(K, Sn∫µr)→H1(K, µr).
(ii) Let sgn : ˆWrsep(K)→K∗/K∗2 be the cohomology map
4.15 Lemma. Letϕ be one of the invariantsper, sgn, det. For separabler-forms Θ and Ψ over K we have
ϕ(Θ⊕Ψ) =ϕ(Θ)·ϕ(Ψ), ϕ(Θ⊗Ψ) =ϕ(Θ)dimΨ·ϕ(Ψ)dimΘ. Proof: The equations hold for the respective maps S
n≥0
Sn∫ µr →K¯∗ on the matrix
algebra.
4.16 Lemma. For an indecomposable separable r-form (L,trL/Khbir) we have (i) per(L,trL/Khbir) =NL/K(b)∈K∗/K∗r. Proof: We use the notation from Lemma 4.7. Under the bijection in Theorem 4.6, the r-form (L,trL/Khbir) corresponds to the class of the cocyle s 7→ B sB−1
(iii) The determinant is the product of the sign and the permanent. The exponents in the formula results from the fact that the cohomology mapK∗/K∗n→K∗/K∗mn induced by the inclusion µn⊂µmn maps the class of a to the class of am.